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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.integration.gauss;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import java.math.MathContext;<a name="line.19"></a>
<FONT color="green">020</FONT>    import java.math.BigDecimal;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.util.Pair;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NotStrictlyPositiveException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.23"></a>
<FONT color="green">024</FONT>    <a name="line.24"></a>
<FONT color="green">025</FONT>    /**<a name="line.25"></a>
<FONT color="green">026</FONT>     * Factory that creates Gauss-type quadrature rule using Legendre polynomials.<a name="line.26"></a>
<FONT color="green">027</FONT>     * In this implementation, the lower and upper bounds of the natural interval<a name="line.27"></a>
<FONT color="green">028</FONT>     * of integration are -1 and 1, respectively.<a name="line.28"></a>
<FONT color="green">029</FONT>     * The Legendre polynomials are evaluated using the recurrence relation<a name="line.29"></a>
<FONT color="green">030</FONT>     * presented in &lt;a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun"<a name="line.30"></a>
<FONT color="green">031</FONT>     * Abramowitz and Stegun, 1964&lt;/a&gt;.<a name="line.31"></a>
<FONT color="green">032</FONT>     *<a name="line.32"></a>
<FONT color="green">033</FONT>     * @since 3.1<a name="line.33"></a>
<FONT color="green">034</FONT>     * @version $Id: LegendreHighPrecisionRuleFactory.java 1374187 2012-08-17 09:50:54Z erans $<a name="line.34"></a>
<FONT color="green">035</FONT>     */<a name="line.35"></a>
<FONT color="green">036</FONT>    public class LegendreHighPrecisionRuleFactory extends BaseRuleFactory&lt;BigDecimal&gt; {<a name="line.36"></a>
<FONT color="green">037</FONT>        /** Settings for enhanced precision computations. */<a name="line.37"></a>
<FONT color="green">038</FONT>        private final MathContext mContext;<a name="line.38"></a>
<FONT color="green">039</FONT>        /** The number {@code 2}. */<a name="line.39"></a>
<FONT color="green">040</FONT>        private final BigDecimal two;<a name="line.40"></a>
<FONT color="green">041</FONT>        /** The number {@code -1}. */<a name="line.41"></a>
<FONT color="green">042</FONT>        private final BigDecimal minusOne;<a name="line.42"></a>
<FONT color="green">043</FONT>        /** The number {@code 0.5}. */<a name="line.43"></a>
<FONT color="green">044</FONT>        private final BigDecimal oneHalf;<a name="line.44"></a>
<FONT color="green">045</FONT>    <a name="line.45"></a>
<FONT color="green">046</FONT>        /**<a name="line.46"></a>
<FONT color="green">047</FONT>         * Default precision is {@link MathContext#DECIMAL128 DECIMAL128}.<a name="line.47"></a>
<FONT color="green">048</FONT>         */<a name="line.48"></a>
<FONT color="green">049</FONT>        public LegendreHighPrecisionRuleFactory() {<a name="line.49"></a>
<FONT color="green">050</FONT>            this(MathContext.DECIMAL128);<a name="line.50"></a>
<FONT color="green">051</FONT>        }<a name="line.51"></a>
<FONT color="green">052</FONT>    <a name="line.52"></a>
<FONT color="green">053</FONT>        /**<a name="line.53"></a>
<FONT color="green">054</FONT>         * @param mContext Precision setting for computing the quadrature rules.<a name="line.54"></a>
<FONT color="green">055</FONT>         */<a name="line.55"></a>
<FONT color="green">056</FONT>        public LegendreHighPrecisionRuleFactory(MathContext mContext) {<a name="line.56"></a>
<FONT color="green">057</FONT>            this.mContext = mContext;<a name="line.57"></a>
<FONT color="green">058</FONT>            two = new BigDecimal("2", mContext);<a name="line.58"></a>
<FONT color="green">059</FONT>            minusOne = new BigDecimal("-1", mContext);<a name="line.59"></a>
<FONT color="green">060</FONT>            oneHalf = new BigDecimal("0.5", mContext);<a name="line.60"></a>
<FONT color="green">061</FONT>        }<a name="line.61"></a>
<FONT color="green">062</FONT>    <a name="line.62"></a>
<FONT color="green">063</FONT>        /**<a name="line.63"></a>
<FONT color="green">064</FONT>         * {@inheritDoc}<a name="line.64"></a>
<FONT color="green">065</FONT>         *<a name="line.65"></a>
<FONT color="green">066</FONT>         * @throws NotStrictlyPositiveException if {@code numberOfPoints &lt; 1}.<a name="line.66"></a>
<FONT color="green">067</FONT>         */<a name="line.67"></a>
<FONT color="green">068</FONT>        @Override<a name="line.68"></a>
<FONT color="green">069</FONT>        protected Pair&lt;BigDecimal[], BigDecimal[]&gt; computeRule(int numberOfPoints) {<a name="line.69"></a>
<FONT color="green">070</FONT>            if (numberOfPoints &lt;= 0) {<a name="line.70"></a>
<FONT color="green">071</FONT>                throw new NotStrictlyPositiveException(LocalizedFormats.NUMBER_OF_POINTS,<a name="line.71"></a>
<FONT color="green">072</FONT>                                                       numberOfPoints);<a name="line.72"></a>
<FONT color="green">073</FONT>            }<a name="line.73"></a>
<FONT color="green">074</FONT>    <a name="line.74"></a>
<FONT color="green">075</FONT>            if (numberOfPoints == 1) {<a name="line.75"></a>
<FONT color="green">076</FONT>                // Break recursion.<a name="line.76"></a>
<FONT color="green">077</FONT>                return new Pair&lt;BigDecimal[], BigDecimal[]&gt;(new BigDecimal[] { BigDecimal.ZERO },<a name="line.77"></a>
<FONT color="green">078</FONT>                                                            new BigDecimal[] { two });<a name="line.78"></a>
<FONT color="green">079</FONT>            }<a name="line.79"></a>
<FONT color="green">080</FONT>    <a name="line.80"></a>
<FONT color="green">081</FONT>            // Get previous rule.<a name="line.81"></a>
<FONT color="green">082</FONT>            // If it has not been computed yet it will trigger a recursive call<a name="line.82"></a>
<FONT color="green">083</FONT>            // to this method.<a name="line.83"></a>
<FONT color="green">084</FONT>            final BigDecimal[] previousPoints = getRuleInternal(numberOfPoints - 1).getFirst();<a name="line.84"></a>
<FONT color="green">085</FONT>    <a name="line.85"></a>
<FONT color="green">086</FONT>            // Compute next rule.<a name="line.86"></a>
<FONT color="green">087</FONT>            final BigDecimal[] points = new BigDecimal[numberOfPoints];<a name="line.87"></a>
<FONT color="green">088</FONT>            final BigDecimal[] weights = new BigDecimal[numberOfPoints];<a name="line.88"></a>
<FONT color="green">089</FONT>    <a name="line.89"></a>
<FONT color="green">090</FONT>            // Find i-th root of P[n+1] by bracketing.<a name="line.90"></a>
<FONT color="green">091</FONT>            final int iMax = numberOfPoints / 2;<a name="line.91"></a>
<FONT color="green">092</FONT>            for (int i = 0; i &lt; iMax; i++) {<a name="line.92"></a>
<FONT color="green">093</FONT>                // Lower-bound of the interval.<a name="line.93"></a>
<FONT color="green">094</FONT>                BigDecimal a = (i == 0) ? minusOne : previousPoints[i - 1];<a name="line.94"></a>
<FONT color="green">095</FONT>                // Upper-bound of the interval.<a name="line.95"></a>
<FONT color="green">096</FONT>                BigDecimal b = (iMax == 1) ? BigDecimal.ONE : previousPoints[i];<a name="line.96"></a>
<FONT color="green">097</FONT>                // P[j-1](a)<a name="line.97"></a>
<FONT color="green">098</FONT>                BigDecimal pma = BigDecimal.ONE;<a name="line.98"></a>
<FONT color="green">099</FONT>                // P[j](a)<a name="line.99"></a>
<FONT color="green">100</FONT>                BigDecimal pa = a;<a name="line.100"></a>
<FONT color="green">101</FONT>                // P[j-1](b)<a name="line.101"></a>
<FONT color="green">102</FONT>                BigDecimal pmb = BigDecimal.ONE;<a name="line.102"></a>
<FONT color="green">103</FONT>                // P[j](b)<a name="line.103"></a>
<FONT color="green">104</FONT>                BigDecimal pb = b;<a name="line.104"></a>
<FONT color="green">105</FONT>                for (int j = 1; j &lt; numberOfPoints; j++) {<a name="line.105"></a>
<FONT color="green">106</FONT>                    final BigDecimal b_two_j_p_1 = new BigDecimal(2 * j + 1, mContext);<a name="line.106"></a>
<FONT color="green">107</FONT>                    final BigDecimal b_j = new BigDecimal(j, mContext);<a name="line.107"></a>
<FONT color="green">108</FONT>                    final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);<a name="line.108"></a>
<FONT color="green">109</FONT>    <a name="line.109"></a>
<FONT color="green">110</FONT>                    // Compute P[j+1](a)<a name="line.110"></a>
<FONT color="green">111</FONT>                    // ppa = ((2 * j + 1) * a * pa - j * pma) / (j + 1);<a name="line.111"></a>
<FONT color="green">112</FONT>    <a name="line.112"></a>
<FONT color="green">113</FONT>                    BigDecimal tmp1 = a.multiply(b_two_j_p_1, mContext);<a name="line.113"></a>
<FONT color="green">114</FONT>                    tmp1 = pa.multiply(tmp1, mContext);<a name="line.114"></a>
<FONT color="green">115</FONT>                    BigDecimal tmp2 = pma.multiply(b_j, mContext);<a name="line.115"></a>
<FONT color="green">116</FONT>                    // P[j+1](a)<a name="line.116"></a>
<FONT color="green">117</FONT>                    BigDecimal ppa = tmp1.subtract(tmp2, mContext);<a name="line.117"></a>
<FONT color="green">118</FONT>                    ppa = ppa.divide(b_j_p_1, mContext);<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>                    // Compute P[j+1](b)<a name="line.120"></a>
<FONT color="green">121</FONT>                    // ppb = ((2 * j + 1) * b * pb - j * pmb) / (j + 1);<a name="line.121"></a>
<FONT color="green">122</FONT>    <a name="line.122"></a>
<FONT color="green">123</FONT>                    tmp1 = b.multiply(b_two_j_p_1, mContext);<a name="line.123"></a>
<FONT color="green">124</FONT>                    tmp1 = pb.multiply(tmp1, mContext);<a name="line.124"></a>
<FONT color="green">125</FONT>                    tmp2 = pmb.multiply(b_j, mContext);<a name="line.125"></a>
<FONT color="green">126</FONT>                    // P[j+1](b)<a name="line.126"></a>
<FONT color="green">127</FONT>                    BigDecimal ppb = tmp1.subtract(tmp2, mContext);<a name="line.127"></a>
<FONT color="green">128</FONT>                    ppb = ppb.divide(b_j_p_1, mContext);<a name="line.128"></a>
<FONT color="green">129</FONT>    <a name="line.129"></a>
<FONT color="green">130</FONT>                    pma = pa;<a name="line.130"></a>
<FONT color="green">131</FONT>                    pa = ppa;<a name="line.131"></a>
<FONT color="green">132</FONT>                    pmb = pb;<a name="line.132"></a>
<FONT color="green">133</FONT>                    pb = ppb;<a name="line.133"></a>
<FONT color="green">134</FONT>                }<a name="line.134"></a>
<FONT color="green">135</FONT>                // Now pa = P[n+1](a), and pma = P[n](a). Same holds for b.<a name="line.135"></a>
<FONT color="green">136</FONT>                // Middle of the interval.<a name="line.136"></a>
<FONT color="green">137</FONT>                BigDecimal c = a.add(b, mContext).multiply(oneHalf, mContext);<a name="line.137"></a>
<FONT color="green">138</FONT>                // P[j-1](c)<a name="line.138"></a>
<FONT color="green">139</FONT>                BigDecimal pmc = BigDecimal.ONE;<a name="line.139"></a>
<FONT color="green">140</FONT>                // P[j](c)<a name="line.140"></a>
<FONT color="green">141</FONT>                BigDecimal pc = c;<a name="line.141"></a>
<FONT color="green">142</FONT>                boolean done = false;<a name="line.142"></a>
<FONT color="green">143</FONT>                while (!done) {<a name="line.143"></a>
<FONT color="green">144</FONT>                    BigDecimal tmp1 = b.subtract(a, mContext);<a name="line.144"></a>
<FONT color="green">145</FONT>                    BigDecimal tmp2 = c.ulp().multiply(BigDecimal.TEN, mContext);<a name="line.145"></a>
<FONT color="green">146</FONT>                    done = tmp1.compareTo(tmp2) &lt;= 0;<a name="line.146"></a>
<FONT color="green">147</FONT>                    pmc = BigDecimal.ONE;<a name="line.147"></a>
<FONT color="green">148</FONT>                    pc = c;<a name="line.148"></a>
<FONT color="green">149</FONT>                    for (int j = 1; j &lt; numberOfPoints; j++) {<a name="line.149"></a>
<FONT color="green">150</FONT>                        final BigDecimal b_two_j_p_1 = new BigDecimal(2 * j + 1, mContext);<a name="line.150"></a>
<FONT color="green">151</FONT>                        final BigDecimal b_j = new BigDecimal(j, mContext);<a name="line.151"></a>
<FONT color="green">152</FONT>                        final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);<a name="line.152"></a>
<FONT color="green">153</FONT>    <a name="line.153"></a>
<FONT color="green">154</FONT>                        // Compute P[j+1](c)<a name="line.154"></a>
<FONT color="green">155</FONT>                        tmp1 = c.multiply(b_two_j_p_1, mContext);<a name="line.155"></a>
<FONT color="green">156</FONT>                        tmp1 = pc.multiply(tmp1, mContext);<a name="line.156"></a>
<FONT color="green">157</FONT>                        tmp2 = pmc.multiply(b_j, mContext);<a name="line.157"></a>
<FONT color="green">158</FONT>                        // P[j+1](c)<a name="line.158"></a>
<FONT color="green">159</FONT>                        BigDecimal ppc = tmp1.subtract(tmp2, mContext);<a name="line.159"></a>
<FONT color="green">160</FONT>                        ppc = ppc.divide(b_j_p_1, mContext);<a name="line.160"></a>
<FONT color="green">161</FONT>    <a name="line.161"></a>
<FONT color="green">162</FONT>                        pmc = pc;<a name="line.162"></a>
<FONT color="green">163</FONT>                        pc = ppc;<a name="line.163"></a>
<FONT color="green">164</FONT>                    }<a name="line.164"></a>
<FONT color="green">165</FONT>                    // Now pc = P[n+1](c) and pmc = P[n](c).<a name="line.165"></a>
<FONT color="green">166</FONT>                    if (!done) {<a name="line.166"></a>
<FONT color="green">167</FONT>                        if (pa.signum() * pc.signum() &lt;= 0) {<a name="line.167"></a>
<FONT color="green">168</FONT>                            b = c;<a name="line.168"></a>
<FONT color="green">169</FONT>                            pmb = pmc;<a name="line.169"></a>
<FONT color="green">170</FONT>                            pb = pc;<a name="line.170"></a>
<FONT color="green">171</FONT>                        } else {<a name="line.171"></a>
<FONT color="green">172</FONT>                            a = c;<a name="line.172"></a>
<FONT color="green">173</FONT>                            pma = pmc;<a name="line.173"></a>
<FONT color="green">174</FONT>                            pa = pc;<a name="line.174"></a>
<FONT color="green">175</FONT>                        }<a name="line.175"></a>
<FONT color="green">176</FONT>                        c = a.add(b, mContext).multiply(oneHalf, mContext);<a name="line.176"></a>
<FONT color="green">177</FONT>                    }<a name="line.177"></a>
<FONT color="green">178</FONT>                }<a name="line.178"></a>
<FONT color="green">179</FONT>                final BigDecimal nP = new BigDecimal(numberOfPoints, mContext);<a name="line.179"></a>
<FONT color="green">180</FONT>                BigDecimal tmp1 = pmc.subtract(c.multiply(pc, mContext), mContext);<a name="line.180"></a>
<FONT color="green">181</FONT>                tmp1 = tmp1.multiply(nP);<a name="line.181"></a>
<FONT color="green">182</FONT>                tmp1 = tmp1.pow(2, mContext);<a name="line.182"></a>
<FONT color="green">183</FONT>                BigDecimal tmp2 = c.pow(2, mContext);<a name="line.183"></a>
<FONT color="green">184</FONT>                tmp2 = BigDecimal.ONE.subtract(tmp2, mContext);<a name="line.184"></a>
<FONT color="green">185</FONT>                tmp2 = tmp2.multiply(two, mContext);<a name="line.185"></a>
<FONT color="green">186</FONT>                tmp2 = tmp2.divide(tmp1, mContext);<a name="line.186"></a>
<FONT color="green">187</FONT>    <a name="line.187"></a>
<FONT color="green">188</FONT>                points[i] = c;<a name="line.188"></a>
<FONT color="green">189</FONT>                weights[i] = tmp2;<a name="line.189"></a>
<FONT color="green">190</FONT>    <a name="line.190"></a>
<FONT color="green">191</FONT>                final int idx = numberOfPoints - i - 1;<a name="line.191"></a>
<FONT color="green">192</FONT>                points[idx] = c.negate(mContext);<a name="line.192"></a>
<FONT color="green">193</FONT>                weights[idx] = tmp2;<a name="line.193"></a>
<FONT color="green">194</FONT>            }<a name="line.194"></a>
<FONT color="green">195</FONT>            // If "numberOfPoints" is odd, 0 is a root.<a name="line.195"></a>
<FONT color="green">196</FONT>            // Note: as written, the test for oddness will work for negative<a name="line.196"></a>
<FONT color="green">197</FONT>            // integers too (although it is not necessary here), preventing<a name="line.197"></a>
<FONT color="green">198</FONT>            // a FindBugs warning.<a name="line.198"></a>
<FONT color="green">199</FONT>            if (numberOfPoints % 2 != 0) {<a name="line.199"></a>
<FONT color="green">200</FONT>                BigDecimal pmc = BigDecimal.ONE;<a name="line.200"></a>
<FONT color="green">201</FONT>                for (int j = 1; j &lt; numberOfPoints; j += 2) {<a name="line.201"></a>
<FONT color="green">202</FONT>                    final BigDecimal b_j = new BigDecimal(j, mContext);<a name="line.202"></a>
<FONT color="green">203</FONT>                    final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);<a name="line.203"></a>
<FONT color="green">204</FONT>    <a name="line.204"></a>
<FONT color="green">205</FONT>                    // pmc = -j * pmc / (j + 1);<a name="line.205"></a>
<FONT color="green">206</FONT>                    pmc = pmc.multiply(b_j, mContext);<a name="line.206"></a>
<FONT color="green">207</FONT>                    pmc = pmc.divide(b_j_p_1, mContext);<a name="line.207"></a>
<FONT color="green">208</FONT>                    pmc = pmc.negate(mContext);<a name="line.208"></a>
<FONT color="green">209</FONT>                }<a name="line.209"></a>
<FONT color="green">210</FONT>    <a name="line.210"></a>
<FONT color="green">211</FONT>                // 2 / pow(numberOfPoints * pmc, 2);<a name="line.211"></a>
<FONT color="green">212</FONT>                final BigDecimal nP = new BigDecimal(numberOfPoints, mContext);<a name="line.212"></a>
<FONT color="green">213</FONT>                BigDecimal tmp1 = pmc.multiply(nP, mContext);<a name="line.213"></a>
<FONT color="green">214</FONT>                tmp1 = tmp1.pow(2, mContext);<a name="line.214"></a>
<FONT color="green">215</FONT>                BigDecimal tmp2 = two.divide(tmp1, mContext);<a name="line.215"></a>
<FONT color="green">216</FONT>    <a name="line.216"></a>
<FONT color="green">217</FONT>                points[iMax] = BigDecimal.ZERO;<a name="line.217"></a>
<FONT color="green">218</FONT>                weights[iMax] = tmp2;<a name="line.218"></a>
<FONT color="green">219</FONT>            }<a name="line.219"></a>
<FONT color="green">220</FONT>    <a name="line.220"></a>
<FONT color="green">221</FONT>            return new Pair&lt;BigDecimal[], BigDecimal[]&gt;(points, weights);<a name="line.221"></a>
<FONT color="green">222</FONT>        }<a name="line.222"></a>
<FONT color="green">223</FONT>    }<a name="line.223"></a>




























































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